Are Transition Maps Implied within an Atlas?

You can simply define the transition maps, once the atlas is given.

There is a transition map which I shall denote $psi_{m,n}$ for every pair of indices $m,n$ having the property that $U_m cap U_n ne emptyset$.

The domain of $psi_{m,n}$ is the set $phi_m(U_m cap U_n) subset mathbb R^k$ (I'm assuming implicitly that $k$ is the dimension of the manifold).

The range (or codomain) of $psi_{m,n}$ is the set $phi_n(U_m cap U_n) subset mathbb R^k$.

And the formula for $psi_{m,n} : phi_m(U_m cap U_n) to phi_n(U_m cap U_n)$ is $$psi_{m,n}(p) = phi_n(phi^{-1}_m(p)), quad p in phi_m(U_m cap U_n) $$

Also, once all of this is written down, one can use the definition of a manifold together with the Invariance of Domain Theorem to prove that the domain and range of $phi_{m,n}$ are both open subsets of $mathbb R^k$, and one can show that $psi_{n,m}$ is an inverse map of $psi_{m,n}$, hence each transition map is a homeomorphism from its domain to its range.

And once that is done, you can now ask yourself questions that are aimed at determining whether your manifold is a $C^infty$ manifold, or a $C^2$ manifold, or a $C^1$ manifold or whatever smoothness property you want. Namely: Are the functions ${psi_{m,n}}$ all $C^infty$? or are they all $C^2$? or $C^1$?